SDS HOMEWORK 3
Jeremy Sapienza & Stefano D’Arrigo
21 Gennaio 2021
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Problem statement
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We are provided with some preprocessed data coming from Autism Brain Image Data Exchange Project. This dataset is partitioned in two subsets: the Autism Spectrum Disorder (ASD) and the Typically Developed (TD) sets.
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As shown in the image above, each subset contains the records of 12 patients. A sample of 145 records is associated to each person. Formally, this sample can be treated as an IID sample defined as:
\[ D^{(i)}_n = \{X_1, ..., X_n\} \sim f_X(\cdot) \text{ for a certain } f_X \text{, having } n=145 \] Each \(X_i\) is defined as a random vector \(X_i = [X_i(1), ..., X_i(D)]^T \in \mathbb{R}^D\), where \(D=116\) is the number of the features, i.e the number of the Regions of Interest (ROI) of the person’s brain. We also assume that each patient within a group can be himself treated as an IID sample from a certain \(f_D(\cdot)\). Thus, moving bottom-up in the data tree, each group formally corresponds to:
\[ D_n = \{D^{(1)}_n, D^{(2)}_n, ..., D^{(12)}_n\} \] It is worth to note that the IID assumptions at the two levels of the data tree are quite crucial. Thanks to them, we have different ways of pooling the data:
- Assume \(D_n = \{X^{(1)}_1, ..., X^{(1)}_n, ..., X^{(12)}_1, ..., X^{(12)}_n\}\), that is ignoring the fact that each \(X^{(i)}\) comes from different people.
- Consider the estimator \(\hat{D}_n = h\big(\{D^{(1)}_n, ..., D^{(12)}_n\}\big)\) for a certain function \(h\).
At this point, we decide to follow the second approach, computing the covariance matrix \(\hat{\Sigma}^{(i)}\) associated to each \(D^{(i)}_n\), \(i=1,...,12\), and taking the mean of them; by the IID property of \(\{D^{(1)}_n, ..., D^{(12)}_n\}\), we have that \(\big\{\hat{\Sigma}^{(i)}\big\}_{i=1}^{12}\) is still IID, being function of each \(D_n\). This choice has many advantages, which will be pointed out further in the discussion.
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Data handling
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After loading the data provided by the professor, create the covariance matrix for each patient; Z-scale this matrix through the Fisher’s Z-transform and after that create a unique matrix per group considering the olympic mean by \(X_n = (x_1, ... , x_{12})\) for each element in the matrix.
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# define a function in order to catch each person and return the correlation matrix between ROIs
get.cor.ROIs <- function(group, person){
frame <- group[[person]]
n <- ncol(frame)
nms <- names(frame)
# takes efficiently the combinations, this is useful for large dataset!
V1 <- rep(nms[1:(n-1)], seq(from=n-1, to = 1, by = -1))
V2 <- unlist(lapply(1:(n-1), function(i)(nms[(i+1):n])))
corr.ROIs <- data.frame(ROI_1=V1, ROI_2=V2) # create the correlation in which I will insert the values
corr.ROIs$z.pearson.corr <- apply(corr.ROIs, 1, function(row) {
atanh(cor(frame[row["ROI_1"]], frame[row["ROI_2"]])) # takes the sets of columns; apply the Z-transform to the
# correlation matrix
})
return(corr.ROIs)
}\[~\]
Now, for each group create for each corresponding patient the matrix of correlations in the Z-scale.
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# create the matrix correlations for all patients
for(person in names(asd_sel)) assign(paste("z.trans.", person, sep=""), get.cor.ROIs(asd_sel, person))
for(person in names(td_sel)) assign(paste("z.trans.", person, sep=""), get.cor.ROIs(td_sel, person))\[~\]
After that, we pooled the data considering the olympic mean between these datasets for each group, this is due to …. ecccccc
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# create a unique matrix for each group
unique.matrix <- function(group) {
mtx <- z.trans.caltech_0051472[c("ROI_1", "ROI_2")] # create matrix with combinations
mtx$olymp.mean <- apply(mtx, 1, function(row) {
values <- c()
for(person in names(group)) {
frame <- get(paste("z.trans.", person, sep="")) # match the address of the string with the real variable
elem <- frame[(frame[["ROI_1"]] == row["ROI_1"]) & (frame[["ROI_2"]] == row["ROI_2"]), "z.pearson.corr"] # select the correlation
values <- c(values, elem)
}
#values <- values[values != min(values)]; values <- values[values != max(values)] # remove the min and max, we have continuous values, so the prob. to have twice the min or max is impossible..
mean(values) # take the mean!
})
return(mtx)
}\[~\]
Now, we call and create the matrix for each group of people as in this way:
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# call the creation of unique matrix
asd_sel.dt <- unique.matrix(asd_sel); head(asd_sel.dt)## ROI_1 ROI_2 olymp.mean
## 1 2001 2002 0.51188381
## 2 2001 2101 0.02486222
## 3 2001 2102 -0.21203874
## 4 2001 2111 -0.05246567
## 5 2001 2112 -0.19052649
## 6 2001 2201 0.14638295td_sel.dt <- unique.matrix(td_sel); head(td_sel.dt)## ROI_1 ROI_2 olymp.mean
## 1 2001 2002 0.416005628
## 2 2001 2101 0.086652336
## 3 2001 2102 -0.334109327
## 4 2001 2111 0.007217533
## 5 2001 2112 -0.150770771
## 6 2001 2201 0.164410598\[~\]
Some plots
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library(ggplot2)
library(hrbrthemes)## NOTE: Either Arial Narrow or Roboto Condensed fonts are required to use these themes.## Please use hrbrthemes::import_roboto_condensed() to install Roboto Condensed and## if Arial Narrow is not on your system, please see https://bit.ly/arialnarrowhalfHeatmap <- function(x) {
# Give extreme colors:
return(ggplot(x, aes(ROI_1, ROI_2, fill = olymp.mean)) +
geom_tile() +
scale_fill_gradient(low="white", high="blue") +
theme_minimal() +
theme(panel.grid = element_blank()))
}
halfHeatmap(asd_sel.dt)
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halfHeatmap(td_sel.dt)
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hist(asd_sel.dt$olymp.mean, probability = T, breaks = 50, col = "orange", main = "Histogram of ASD group", xlab = "olympic mean", ylab = "count", border = "white")
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hist(td_sel.dt$olymp.mean, probability = T, breaks = 50, col = "steelblue", main = "Histogram of TD group", xlab = "olympic mean", ylab = "count", border = "white")
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hist(rbind(asd_sel.dt[["olymp.mean"]], td_sel.dt[["olymp.mean"]]), probability = T, breaks = 50, col = "purple", main = "Histogram of ASD+TD group", xlab = "olympic mean", ylab = "count", border = "white")
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Association graphs
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Start to find the quantile value for these two groups. Since working on the Z-scale, the threshold is also in the Z-scale.
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z.t <- apply(rbind(asd_sel.dt["olymp.mean"], td_sel.dt["olymp.mean"]), 2, quantile, probs=0.8)
paste("the threshold at the 80th percentile for the subjects is: ", round(as.numeric(z.t), 3), sep = "")## [1] "the threshold at the 80th percentile for the subjects is: 0,125"\[~\]
We compute the confidence interval for \(\bar{Z}_{12}(j,k)\). Remember that we started computing the estimate of the covariance matrix for each person in each of the two groups, having:
\[ \Big\{\hat{\rho}^{(i)}_{j,k}\Big\}_{i=1}^{12}, \text{ having } j,k \in \{1,...,116\}, \text{ } j\neq k \]
Then, we applied the Fisher’s Z-transform:
\[ \Big\{\hat{Z}^{(i)}_{j,k}\Big\}_{i=1}^{12} = \Big\{atan\left(\hat{\rho}^{(i)}_{j,k}\right)\Big\}_{i=1}^{12} \]
Finally, we computed the estimator:
\[ \overline{Z}_{12}(j,k) = \frac{1}{12}\sum_{i=1}^{12}\hat{Z}^{(i)}_{j,k} \text{ ,} \]
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remembering that \(\hat{Z}^{(i)}_{j,k}\) are \(IID\), being function of \(\hat{\rho}^{(i)}_{j,k}\), respectively.
Now, we calculate the confidence interval for each \(\overline{Z}_{12}(j,k)\). Starting from:
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\[ \frac{\overline{Z}_{12}(j,k) - z_{j,k}}{\sigma_{j,k} / \sqrt{12}}, \\ \text{ where } \sigma_{j,k} = \frac{1}{\sqrt{145 - 3}} \]
and applying the Bonferroni’s correction:
\[ \frac{\alpha}{m}, \text{ where } m = \begin{pmatrix} D \\ 2 \end{pmatrix}, \text{ } D=116 \]
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we end up with:
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\[ C_{12}^{Z(j,k)}\Big(\frac{\alpha}{m}\Big) = \bigg[\overline{Z}_{12}(j,k) \mp z_{\alpha/2m} \cdot \frac{\sigma_{j,k}}{\sqrt{12}}\bigg] \]
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conf.int <- function(dt, m = 1) { # m : Family-Wise Error Rate correction (Bonferroni)
# Asymptotic variance
n.samp <- 145 # number of IID samples on which we computed the covariance matrix
sigma <- 1 / sqrt(n.samp - 3) # standard deviation of the (j,k)-th Z-transformed Pearson coefficent
n.p <- 12 # number of people in each of the two groups
se <- sigma / sqrt(n.p) # standard error of the estimator Z12
# Confidence interval - Setup
alpha <- 0.05 # confidence level WITHOUT the Bonferroni's correction
# Confidence interval
cint <- sapply(dt$olymp.mean, function(z.i) {
list(confidence_interval = c(lb = z.i - se * qnorm(1 - alpha / (2*m)),
ub = z.i + se * qnorm(1 - alpha / (2*m))))
})
return(cint)
}\[~\]
Finally:
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## Compute the confidence interval
m <- (116 * 115) / 2 # number of intervals
asd_sel.dt$cint <- conf.int(asd_sel.dt, m); asd_sel.dt$cint[1:3]## $confidence_interval
## lb ub
## 0,4033775 0,6203901
##
## $confidence_interval
## lb ub
## -0,08364405 0,13336849
##
## $confidence_interval
## lb ub
## -0,3205450 -0,1035325td_sel.dt$cint <- conf.int(td_sel.dt, m); td_sel.dt$cint[1:3]## $confidence_interval
## lb ub
## 0,3074994 0,5245119
##
## $confidence_interval
## lb ub
## -0,02185393 0,19515860
##
## $confidence_interval
## lb ub
## -0,4426156 -0,2256031\[~\]
Show an example of confidence interval between two ROIs:
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plotCint <- function(ROI.1, ROI.2, dt) {
hist(dt$olymp.mean, probability = T, breaks = 50, col = "pink", main = "Interval of a data point", xlab = "olympic mean", ylab = "count", border = "deeppink")
corresponding.interval <- function(ROI.1, ROI.2, group) {
cint <- group[(group$ROI_1 == ROI.1) & (group$ROI_2 == ROI.2), "cint"]
lb <- cint[[1]]["lb"]; ub <- cint[[1]]["ub"]
estimate.corr <- group[(group$ROI_1 == ROI.1) & (group$ROI_2 == ROI.2), "olymp.mean"]
info <- list("lb" = lb, "ub" = ub, "olymp.mean" = estimate.corr)
return(info)
}
cint <- corresponding.interval(ROI.1 = ROI.1, ROI.2 = ROI.2, dt)
rect(cint$lb, -10, cint$ub, +10, border = NA, col = viridis::viridis(1, .3))
points(cint$olymp.mean, 0, col = "green", pch = 19)
abline(v = cint$olymp.mean, lwd = 3, col = viridis::viridis(1, .1))
text(cint$olymp.mean, 2, expression(hat(p)[olymp]), pos = 4)
}
plotCint(ROI.1 = "2001", ROI.2 = "2002", asd_sel.dt)
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plotCint(ROI.1 = "2001", ROI.2 = "2002", td_sel.dt)
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We are now ready to estimate the adjacency matrix of the association graphs:
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## Estimate the adjacency matrix given the Z-transformed Pearson correlation coefficient
get.est.adj.mat <- function(dt, dec.rule, t) {
# create the adj matrix
nameVals <- sort(unique(unlist(dt[1:2]))) # set up storage matrix, get names for row and columns
# construct 0 matrix of correct dimensions with row and column names
adj_mat <- matrix(0, length(nameVals), length(nameVals), dimnames = list(nameVals, nameVals))
# fill in the matrix with matrix indexing on row and column names
adj_mat[as.matrix(dt[c("ROI_1", "ROI_2")])] <- 0
# put edge according to the decision rule
# TODO: SUBSTITUTE THE FOR LOOP WITH A MORE EFFICIENT SOLUTION
for(idx in rownames(dt)){
if( dec.rule(dt, idx, t) ) {
adj_mat[as.character(dt[idx, "ROI_1"]), as.character(dt[idx, "ROI_2"])] <- 1
}
}
return(adj_mat)
}
## check if the two intervals int1 and int2 are overlapping
are.joint <- function(int1, int2) return((int1[1] <= int2[2]) && (int2[1] <= int1[2]))
## check the simple threshold condition
check.threshold <- function(dt, idx, t) {
val <- abs(dt[idx, "olymp.mean"])
return(val >= t)
}
## check the confidence interval condition
check.conf.int <- function(dt, idx, t) {
c.int <- dt[idx, "cint"]$confidence_interval
t.int <- c(-t, t)
return(!are.joint(c.int, t.int))
}\[~\]
The association graph taking into account the confidence interval with the Bonferroni’s correction is:
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require(igraph, quietly = TRUE)##
## Attaching package: 'igraph'## The following objects are masked from 'package:stats':
##
## decompose, spectrum## The following object is masked from 'package:base':
##
## union# Adjacency matrix of the estimated graph with the Bonferroni's correction
adj_mat_asd_bonf <- get.est.adj.mat(asd_sel.dt, check.conf.int, z.t)
adj_mat_td_bonf <- get.est.adj.mat(td_sel.dt, check.conf.int, z.t)
# Estimated graph
G.asd.bonf <- graph_from_adjacency_matrix(adj_mat_asd_bonf, mode = "undirected")
G.td.bonf <- graph_from_adjacency_matrix(adj_mat_td_bonf, mode = "undirected")\[~\]
Plot the graphs with the threshold, with Bonferroni’s multiplicity adjustment:
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# To plot the correlation graph(s)
plot(G.asd.bonf, vertex.size = 4, edge.width = .3, vertex.color = "orchid", label.col = "black",
main = "ASD Association Graph", sub = "(Bonferroni's multiplicity adjustment)",
layout = layout.kamada.kawai, vertex.label.cex = 0.7, vertex.label.dist = .8)
plot(G.td.bonf, vertex.size = 4, edge.width = .3, vertex.color = "dodgerblue", label.col = "black",
main = "TD Association Graph", sub = "(Bonferroni's multiplicity adjustment)",
layout = layout.kamada.kawai, vertex.label.cex = 0.7, vertex.label.dist = .8)
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The association graph taking into only the threshold is:
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# Adjacency matrix of the estimated graph with the Bonferroni's correction
adj_mat_asd_thr <- get.est.adj.mat(asd_sel.dt, check.threshold, z.t)
adj_mat_td_thr <- get.est.adj.mat(td_sel.dt, check.threshold, z.t)
# Estimated graph
G.asd.thr <- graph_from_adjacency_matrix(adj_mat_asd_thr, mode = "undirected")
G.td.thr <- graph_from_adjacency_matrix(adj_mat_td_thr, mode = "undirected")\[~\]
Plot the graphs with the threshold, without adjustment:
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plot(G.asd.thr, vertex.size = 4, edge.width = .3, vertex.color = "orchid", label.col = "black",
main = "TD Association Graph", sub = "(Threshold only)",
layout = layout.kamada.kawai, vertex.label.cex = 0.7, vertex.label.dist = .8)
plot(G.td.thr, vertex.size = 4, edge.width = .3, vertex.color = "dodgerblue", label.col = "black",
main = "TD Association Graph", sub = "(Threshold only)",
layout = layout.kamada.kawai, vertex.label.cex = 0.7, vertex.label.dist = .8)
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The association graph taking into account the confidence interval without the Bonferroni’s correction is:
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# Recompute the confidence intervals without the multiplicity adjustment
asd_sel.dt$cint <- conf.int(asd_sel.dt, m = 1)
td_sel.dt$cint <- conf.int(td_sel.dt, m = 1)
# Adjacency matrix of the estimated graph with the Bonferroni's correction
adj_mat_asd_no_bonf <- get.est.adj.mat(asd_sel.dt, check.conf.int, z.t)
adj_mat_td_no_bonf <- get.est.adj.mat(td_sel.dt, check.conf.int, z.t)
# Estimated graph
G.asd.no.bonf <- graph_from_adjacency_matrix(adj_mat_asd_no_bonf, mode = "undirected")
G.td.no.bonf <- graph_from_adjacency_matrix(adj_mat_td_no_bonf, mode = "undirected")\[~\]
Plot the graphs without Bonferroni’s multiplicity adjustment:
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plot(G.asd.no.bonf, vertex.size = 4, edge.width = .3, vertex.color = "orchid", label.col = "black",
main = "TD Association Graph", sub = "(Without Bonferroni's multiplicity adjustment)",
layout = layout.kamada.kawai, vertex.label.cex = 0.7, vertex.label.dist = .8)